Kahler Gradient Ricci Solitons (KGRS) are fundamental in the theory of Ricci flows. This talk's focus is on complex dimension two and will first review the recent beautiful classification of the shrinking possibly non-compact case. Then we'll discuss an approach to detect symmetry applicable to all cases. This differs from and should complement the popular perspective based on asymptotic behaviors. The idea is inspired by Morse-theoretic aspects of symplectic geometry and involves the understanding of singular sets of a moment map. Precisely, we'll show that a KGRS in complex dimension two is an integrable Hamiltonian system; if the system is generic, then it admits a holomorphic 2-torus action.

A central theme in Kähler geometry is the search for canonical Kähler metrics, such as Kähler-Einstein metrics, constant scalar curvature Kähler (cscK for short) metrics, extremal metrics, Kähler-Ricci solutions, etc. The concept of weighted cscK metrics, introduced by Lahdili in 2019, provides a unification of all the above geometric settings. In this talk I will give a panorama of what it is known about these metrics and I will present a criteria for ensuring their existence. This is a joint work with S. Jubert and A. Lahdili.

Let P be a homogeneous polynomial in N+1 complex variables of degree d. The logarithmic Mahler Measure of P (denoted by m(P) ) is the integral of log|P| over the sphere in C^{N+1} with respect to the usual Hermitian metric and measure on the sphere.  Now let X be a smooth variety embedded in CP^N by a high power of an ample line bundle and let $\Delta$ denote a generalized discriminant of X wrt the given embedding , then $\Delta$ is an irreducible homogeneous polynomial in the appropriate space of variables.  In this talk I will discuss work in progress whose aim is to find an asymptotic expansion of m(\Delta) in terms of elementary functions of the degree of the embedding.  

We will discuss the isotopy problem of symplectic forms in a fixed symplectic class on compact four manifolds.

We will introduce a nonlinear Hodge flow for a general approach. We will also discuss the hypersymplectic flow, introduced by Fine-Yao to study hypersymplectic four manifolds.

Abstract: We present a uniform description of  SU(3) structures in dimension 6 as well as  G_2 structures in dimension 7 in terms of a characterising spinor and the spinorial field equations it satisfies. We apply the results to hypersurface theory to obtain new embedding theorems, and give a general recipe for building conical manifolds. The approach sheds new light on connections with torsion and their invariants.